= Compactly generated space
A compactly generated space, in the k-space convention used here, is a <topological space> whose closed subsets are detected by continuous maps from compact Hausdorff spaces. A subset $A\subseteq X$ is closed exactly when $f^{-1}(A)$ is closed for every continuous $f:K\to X$ with $K$ compact Hausdorff.
Every metric space has this property. If $A$ is not closed, choose $x_n\in A$ tending to $x\notin A$. The map from $K=\{0\}\cup\{1/n:n\geq1\}$ sending $0$ to $x$ and $1/n$ to $x_n$ is continuous, but the inverse image of $A$ is not closed. A compact test therefore detects the failure.
The <kification> $kX$ imposes exactly these compact-test closed sets on the same underlying set. <Compactly generated products> use $k(X\times Y)$ so that products remain inside the chosen category. Some authors additionally require the weak Hausdorff property when saying compactly generated; that is an extra convention, used in the cited lecture notes.<Compactly Generated Spaces (18.906 Lecture 2)>[Definitions1.3–1.4 and section2]{ref}
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